【正文】 trol effectors. The free bodydiagram (FBD) for the vehicle under consideration is shown inFig. 7. The body fixed reference frame is labelled on the FBD with itsorigin atthe vehicle’s centre of gravity. The z axis is pointing up, the0,540,570,600,630,660,69 NumericAnalyticZL (m)048100,51Time (s)01530456075NumericAnalyticLateral pressure force (KN)2604810Time (s)26cornering manoeuvre.0 100,40,30,20,10,00,10,20,30,40,5NumericAnalyticTime (s)YL (m)ZL (m)0,510,540,570,600,630,660,69 NumericAnalytic14Numeric1032 468 010Time (s)2468laneM. Toumi et al. / European Journal of Mechanics A/Solids 28 (2020) 1026–10341030x axis is pointing towards the front of the vehicle, and the y axis ispointing towards the vehicle’s right side. This model assumes thatleft and right steer angles are the same for front and rear axles.8910111213AnalyticIxx (kg m2)010Time (s)2468Fig. 6. Response to singleThis model divides the vehicle’s mass into unsprung mass, muand sprung mass, ms. The sprung mass is the body of the vehiclesupported by its suspension. This sprung mass is assumed to pivotabout an imaginary roll axis. Neglecting the pitch and verticalmovement, the equations are derived from the vehicle’s linear andangular movements:Xi188。 z222。x。 w C2 V 254。0。xu。zs222。254。CrSrcos240。 FzrC0DFzr(15)where R, L, f and r represent right, left, front and rear side respectively. Fzis the vertical wheel load。FzR.External forces acting on the vehicle are gravitational forces,tireroad contact forces, and suspensions forces. The externalmoments are the moments due to tireroad forces and suspensions0246810012345678910Time (s)Time (s)1086420Yaw rate (deg/s)0,000,050,100,150,200,250,30(Rigid)(Liquid)Lateral rear load transfertFig. 10. Vehicle response to single0123456789106420246Roll angle (deg)Time (s)(Rigid)(Liquid)2468(Rigid)(Liquid)A/Solids 28 (2020) 1026–1034forces. In this case we assume that the vehicle travels at a fixedspeed, so the longitudinal force will be negligible.The nonlinear variation of tire cornering stiffness with verticalload is described using the quadratic equation to obtain the lateraltire force Fy:Fyi188。d。 d is steer angle。 so lateral load transfer is significant even atmodest levels of lateral acceleration. This difference is moresignificant for the vehicle with a liquid charge than with a rigidcharge.7. ConclusionThe results achieved in this study clearly demonstrate theperformance of this simplified analytical algorithm to characterizethe sloshing liquid in a road tanker. The parison between thisand a plex numerical model shows a good correlation for casesof difficult movement such as steadystate turning and both singleand double lane change manoeuvres. The small difference betweenthe two models is probably due to the assumption of linearity andthe iterative calculation used to capture the form of the freesurfacefor the analytical model pared to the numerical model. In thesecond simulation, the analytical model is coupled with a fullnonlinear unit vehicle. The parison with a vehicle transportinga rigid load shows that the forces and moments arising froma directional manoeuvre yield a considerable dynamic load shift inthe roll and pitch planes due to the sloshi。 right222。wuC2 ruijC1yC0V 254。1, then the right or left wheels lift off and therollover coefficient (lateral load transfer) takes on the value LTR 188。DFzrFzLf188。f222。12hKfSfsin240。xs。 ru188。C18 C19TC18 C19TTime (s)change manoeuvre.453015015304560Lateral pressure force (KN)NumericAnalytic0102 4 6 8and : wu188。 (12)Fig. 7. Free body diagram of the vehicle model.jvtj j j j j jwhere : a 188。wjC2Ijwj。x。 m) is partially filled (50%) with domestic oil (r188。 2。x2kC17Vc。 F!(9)coxzyFcrcFig. 2. Force and moment induced by liquid pressure.M. Toumi et al. / European Journal of Mechanics A/Solids 28 (2020) 1026–10341028pared to that of the liquid. In this direction, the movement ofthe liquid is independent, or free. The only influence gas has is thepressure it exerts on the surface of the liquid. In other words, thegas–liquid surface is not forced, but free. Several numericalapproaches have been employed tomodel the multiphase problem.The most popular are:C15 Combined Lagrangian Eulerian (ALE): In this formulation themesh partly moves and bees deformed because it followsthe material (Lagrangian formulation). At the same time thematerial can also cross the mesh (Eulerien formulation) (Geuzaine et al., 2020。254。 m 188。1。g as shown in Fig. 1.The shift in the centre of the mass and the variation of theinertia matrix is calculated according to the volume integral, by thefollowing equation as:xi188。C01rVP 254。 l represents the factor ofvolume pression and F is the external body force.To develop the analytical model, one needs to apply someassumptions. In this study we assume that the liquid is inpressible (l188。V240。0 (1)For viscous and Newtonian liquids the momentum equation isobtained by:rC18vVvt254。 (4)Starting from Eq. (4), the total derivative of the pressure will havethe following form:dP 188。Fr。is toLiquidtheusuati. Thegoodpaperreddisturbancerigid containersfew decades.andonater oscillationpoolsto both highway safety and the environment (Botkin,1970).Tank trucks employed in general purpose chemical transportation encounter partial fill conditions due to the varyingweight density of the products and the laws governing axle loads,while those employed in fuel transportation encounter partial fillApart fr